Astle reached his 16th limited-overs international hundred and his first against the West Indies when he took a single off the first ball of Chris Gayle's final over.
The 34-year-old got himself back on strike for the last three deliveries and smacked them all over the boundary rope to finish unbeaten on 118 from 126 balls.
Hamish Marshall, recalled to the team for Jamie How, chipped in with a timely 43 off 54 balls in a record 99-run partnership with Astle for the fifth wicket.
The West Indies, beaten in the first two games of the five-match series, looked to have gained the upper hand when fast bowler Ian Bradshaw ripped through the top order.
The lively paceman dismissed Lou Vincent (25), New Zealand skipper Stephen Fleming (15) and Scott Styris for one as the home team crashed to 64-3 by the 15th over.
The hosts then slumped to 87-4 when Peter Fulton fell to Dwayne Smith for 21, bringing Marshall and Astle together, who set about putting their team firmly back in control of the match.
The pair broke the 10-year-old record, set by Fleming and Chris Cairns, for the highest fifth wicket partnership by a New Zealand pair against the West Indies.